Hankel matrix

I. is the identity matrix and z is defined by z (k-v)AT), is determined. In the second step, this model is transformed into a discrete-time state space model. This is achieved by making an approximate realization of the markov parameters (the impulse responses) of the ARX model ( ). The order of the state space model is determined by an evaluation of the singular values of the Hankel matrix (12.). 

Using Eq. (45), which states that the n x n Hankel matrix H (cs) is equal to the matrix LU of the evolution operator U raised to the power s, namely Us,... 

The data used in subspace state-space model development consists of the time series data of output and input variables. For illustration, assume a case with only output data and the objective is to build a model of the form Eq. 4.62. Since the whole data set is already known, it can be partitioned as past and future with respect to any sampling time. Defining a past data window of length K and a future data window of length J that are shifted from the beginning to the end of the data set, stacked vectors of data are formed. The Hankel matrix (Eq. 4.64) is used to develop subspace... 

Canonical variate realization requires that covariances of future and past stacked observations be conditioned against any singularities by taking their square roots. The Hankel matrix is scaled by using and Rj... 

Many scientific computations (for signal processing, for partial differential equations, in statistics, for approximation of functions by polynomials, and so on) are reduced to solving Toeplitz or Hankel systems Ax = b, having Toeplitz or Hankel matrix A. A Hankel matrix becomes a Toeplitz matrix by appropriate row interchange (reflecting the rows about the median row), so we shall consider only Toeplitz systems. Anmxn Toeplitz matrix is defined by its first row and its first column, which requires only m + n 1 units of storage space. 

Sanchez-Gasca J.J., Chow J.H., 1997. Computation of power system low-order models Irom time domain simulation using a Hankel matrix, IEEE Trans. Power Systems, Vol. 12, No. 4,pp. 1461-1467. 

Eq. (12) implies that of the m = 64 singular values of the Hankel matrix H only two large ones will emerge above 62 small and almost equal ones. This fact can be illustrated with a typical plot of the singular values obtained from a data set for a monodisperse sample (figure 1). 

Figure 1 Singular values of the Hankel matrix of the intensity autocorrelation function of a 220 nm diameter monodisperse latex, sampled at 128 equidistant time windows at a single angle of 90° (semi-logarithmic scale).

In Eq. 2, the block Hankel matrix of the measurements is subdivided into two sub-matrices, named as Yp and Yf, which are usually referred to as past and future output block matrices. The orthogonal projection of the row space of Yf onto the row space of Yp can be directly calculated yielding to matrix Pi, which is known as projection matrix. The observability matrix Oi of the system - which, in turn, allows to compute A and C - can be estimated from the SVD of matrix WiPiW2, where Wi and W2 are convenient weight matrices. The available SSI-data algorithms essentially differ for the expressimis of the weight matrices (van Overschee and De Moor 1996). 

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